Assume that $mathbb{D}$ is the open unit disk. Applying Ozaki's conditions, we consider two classes of locally univalent, which denote by $mathcal{G}(alpha)$ and $mathcal{F}(mu)$ as follows begin{equation*}  mathcal{G}(alpha):=left{fin mathcal{A}:mathfrak{Re}left( 1+frac{zf^{prime prime }(z)}{f^{prime }(z)}right) <1+frac{alpha }{2},quad 0<alphaleq1right}, end{equation*} and begin{equation*}  ma...

The Furstenberg–Sárközy theorem asserts that the difference set E?E of a subset E?N with positive upper density intersects image any polynomial P?Z[n] for which P(0)=0. Furstenberg's approach relies on correspondence principle and version Poincaré recurrence theorem, is derived from ergodic-theoretic result measure-preserving system (X,B,?,T) A?B ?(A)>0, one has c(A):=limN???1N?n=1N?(A?T?P(n)A)...

The long persistent discrepancies in $b\,\to\,s\, \ell\, \ell$ quark level transitions continue to be the ideal platform for an indirect search of new physics that lies beyond SM. measurements $R_K$, $R_{K^*}$, $P_5^{\prime}$, $\mathcal{B}(B_s\, \to\, \phi\, \mu^+\,\mu^-)$ and persistently deviate from standard model expectations. Similarly, tests lepton flavor universality performed using isos...

A finite abstract simplicial complex G defines two finite simple graphs: the Barycentric refinement G1, connecting two simplices if one is a subset of the other and the connection graph G′, connecting two simplices if they intersect. We prove that the Poincaré-Hopf value i(x) = 1−χ(S(x)), where χ(S(x)) is the Euler characteristics of the unit sphere S(x) of a vertex x in G1, agrees with the Gre...

A set of integers A is said to tile the integers if there is a set C ⊂ Z such that every integer n can be written in a unique way as n = a + c with a ∈ A and c ∈ C. Throughout this paper we will assume that A is finite. It is well known (see [7]) that any tiling of Z by a finite set A must be periodic: C = B + MZ for some finite set B ⊂ Z such that |A| |B| = M . W then write A ⊕ B = Z/MZ. Newma...

A set of integers A is said to tile the integers if there is a set C ⊂ Z such that every integer n can be written in a unique way as n = a+ c with a ∈ A and c ∈ C. Throughout this paper we will assume that A is finite. It is well known (see [7]) that any tiling of Z by a finite set A must be periodic: C = B + MZ for some finite set B ⊂ Z such that |A| |B| = M . We then write A⊕B = Z/MZ. Newman ...